Two trace inequalities for operator functions

In this paper we show that for a non-negative operator monotone function $f$ on $[0,\mathrm{\infty })$ such that $f(0)=0$ and for any positive semidefinite matrices $A$ and $B$, $$Tr((A-B)(f(A)-f(B)))\le Tr(\mathrm{\mid }A-B\mathrm{\mid }f(\mathrm{\mid }A-B\mathrm{\mid }))\mathrm{.}$$ When the function $f$ is operator convex on $[0,\mathrm{\infty })$, the inequality is reversed.